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AS Mathematics- Core Mathematics 2
Chapter 1: Algebra and Functions
Dividing a polynomial by ( x ± p ):

x2
x - 3 x 3 + 2x 2 -17x + 6
-x

3

- 3x
5x 2 -17x

x 2 + 5x
x - 3 x 3 + 2x 2 -17x + 6
-x

3

- 3x
5x 2 -17x
2
- 5x -15x
-2x + 6

x 2 + 5x - 2
x - 3 x 3 + 2x 2 -17x + 6
-x

3

- 3x
5x 2 -17x
2
- 5x -15x
-2x + 6
- -2x + 6
0 (no remainder)
...

3

2

2

Factorising a polynomial by using the factor theorem:
-If f(x) is a polynomial and
e
...
Factorise

b
f ( ) = 0 then (ax-b) is a factor of f(x)
...


x ´ ax k = 2x 3 so a=2 and k=2
...

Therefore:

(x - 3)(2x 2 + 7x + 3) = 2x 3 + x 2 -18x - 9
(x - 3)(2x +1)(x + 3) = 2x 3 + x 2 -18x - 9
Finding the remainder using the remainder theorem:
-If a polynomial f(x) is divided by (ax-b) then the remainder is
Find the remained when
theorem:

b
f ( )
...


Chapter 2: The Sine and Cosine Rule
The sine rule:
-You can use the sine rule to find an unknown length when you know two angles and
one of the opposite sides:

a
b
c
=
=
sin A sin B sinC
-You can use the sine rule to find an unknown angle in a triangle when you know the
two sides and one of their opposite angles:

sin A sin B sinC
=
=
a
b
c
-You can sometimes find two solutions for a missing angle:
 When the angle you are finding is larger than the given angle, there are two
possible results
...

 In general, sin(180-x)°=sinx°, e
...
sin30°=sin150°
...

-Used to find an unknown angle in a triangle if you know the lengths of all the sides
Area of a triangle:

1
Area of a triangle = absinC
2

-Used if you know the length of two sides (a and b) and the value of the angle C
between them
...

loga 1= 0 ( a > 0 )
loga a =1 ( a > 0 )

Laws of logarithms:

loga x = log x
loga xy = loga x + loga y

x
log a ( ) = loga x - loga y
y
loga (x)k = k loga x
1
loga ( ) = -log a x
x
Solving equations
You can solve an equation such as
each side)
...


Chapter 4: Coordinate geometry in the (x,y) plane

log b x
log b a
...


-The angle in a semicircle is a right angle
...


-The angle between a tangent and a radius is 90
...


Factorial notation:

n! = n ´(n -1)´(n - 2)´(n -3)´3´ 2 ´1
4!= 4 ´3´ 2 ´1= 24
0! =1

-The number of ways of choosing ‘r’ items from a group of ‘n’ items is written as

æ n ö
n!
÷ and is calculated by
Cr or ç
(n - r)!r!
...
g
...
= +n Cn bn
Similarly:

(a + bx)n =n C0 an +n C1an-1bx +n C2 an-2b2 x 2 +n C3an-3b3 x3 +
...

If the arc AB has length r, then the angle AOB is 1 radian (1c or 1 rad)
...

π radians = 180°
...


l = rq

Calculating the area of a sector of a circle:
-It is simpler with radians
...


1
A = r 2 (q - sinq )
2

Chapter 7: Geometric Sequences and Series

Geometric Sequences:
-To get from one term to he next we multiply by the same number each time
...

-You can define a geometric sequence using the first term ‘a’ and the common ratio
‘r’
...


ar 3,
...

Sum of geometric series:

a(1- r n )
Sn =
1- r
Sum to infinity of a convergent geometric series:

S¥ =

a
, if r <1
1- r

Chapter 8: Graphs of trigonometric functions

Trigonometric ratios:
opposite
adjacent
opposite
sin q =
cosq =
tanq =
hypotenuse
hypotenuse
adjacent

Definitions of sinθ, cosθ and tanθ for all values of θ:
y
x
y
sin q = cosq = tanq =
r
r
x
Where ‘x’ and ‘y’ are the coordinates of ‘P’ and ‘r’ is the length of OP
...

In the second quadrant only sin is +ve
...

In the fourth quadrant cos is +ve
...

b
Amplitude (a)= 1
...

Cos graph:
-Graph:

y = acos(bx)
y = cos x
360
Period (
) = 360°
...

-Rule for finding the other cos x angle:

360-θ
...

b
Amplitude (a)= 1
...

Sketching simple transformations on the graphs of sinθ, cosθ and tanθ:
-LOOK AT CHAPTER ON SKETCHING GRAPHS
...

-For an increasing function f(x) in the interval (a,b) f’(x)<0 in the interval a £ x £ b
...

-For f(x) to be an decreasing function f’(x)<0 (gradient is negative)
...
It may be minimum, maximum or a
point of inflection:
 The points where f(x) stops increasing and begins to decrease are called
maximum points (gradient is positive, 0 and then negative)
...

 The points where f(x) increases twice or decreases twice are points of
inflexion (gradient is positive, 0 and then positive or gradient is negative, 0
and then negative)
...
You can use
this to find the range of the function
...

dx
 Substitute the value(s) of x, which you have found, into the equation y=f(x) to
find the corresponding value(s) of y
...

-You can also find out whether something is a maximum, minimum or point of

d2y
d 3y
inflection by finding the value of
and, where necessary,
at the stationary
dx 2
dx 3
point(s)
...

If
= 0 and
dx 2
dx



d y
dy
< 0 , the point is a minimum point
...

2



dy
dy
¹ 0 , the point is a point of inflexion
...

 A second solution is (180°-α)
...

-For cosx=k:
 A first solution is your calculator value ( a = cos-1 k )
...

 Other solutions are found by adding or subtracting multiples of 360°
...

 A second solution is (180°+α)
...

Solving other trogonmetrical equations where sin(nθ+α)=k, cos(nθ+α)=k and
tan(nθ+α)=p:
-You replace nθ by X so that equation reduces to the type you have solved before
...


Chapter 11: Integration

Definite integral:
b

b

a

a

ò f '(x)dx = [ f (x)]

= f (b) - f (a) , provided that f’ is the derived function of f
...

a

Areas of curves under the x-axis:
b

-If the area between a curve and the x-axis lies below the x-axis then

ò y dx will give
a

you a negative answer
...


Area between a curve and a straight line:
-You find the area under the straight line and minus it by the area under the curve or
vice versa
...
Each strip
a

will be of width h, so h =

b-a

...


-Finally you join adjacent points to form n trapeziums and approximate the original
area by the sum of the areas of these n trapeziums:

1
Area of trapezium = (y0 + y1 )h
2

-Finding area under the curve:
b

ò y dx » 2 h(y

1
1
+ y1 ) + h(y1 + y2 ) +
...
+ yn-1 + yn-1 + yn )

1

0

a
b

1

0

a
b

ò y dx » 2 h [ y
1

0

a

+ 2(y1 + y2 +
...

n



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